Procedure: Generate an optimised Latin hypercube design¶
Description and Background¶
A Latin hypercube (LHC) is a random set of points in \([0,1]^p\) constructed so that for \(i=1,2,\ldots,p\) the i-th coordinates of the points are spread evenly along [0,1]. However, a single random LHC will rarely have good enough space-filling properties in the whole of \([0,1]^p\) or satisfy other desirable criteria. Therefore, it is usual to generate many LHCs and then select the one having the best value of a suitable criterion.
We present here the procedure for a general optimality criterion and also include details of the most popular criterion, known as maximin. A thorough consideration of optimality criteria may be found in the discussion page on technical issues in training sample design for the core problem (DiscCoreDesign).
Inputs¶
- Number of dimensions \(p\)
- Number of points desired \(n\)
- Number of LHCs to be generated \(N\)
- Optimality criterion \(C(D)\)
Outputs¶
- Optimised LHC design \(D = \{x_1, x_2, \ldots, x_n\}\)
Procedure¶
Procedure for a general criterion¶
- For \(k=1,2,\ldots,N\) independently, generate a random LHC \(D_k\) using the procedure page on generating a Latin hypercube (ProcLHC) and evaluate the criterion \(C_k = C(D_k)\).
- Let \(K=\arg\max\{D_k\}\) (i.e. \(K\) is the number of the LHC with the highest criterion value).
- Set \(D=D_K\).
Note that the second step here assumes that high values of the criterion are desirable. If low values are desirable, then change argmax to argmin.
Maximin criterion¶
A commonly used criterion is
where \(|x_j - x_{j^\prime}|\) denotes a measure of distance between the two points \(x_j\) and \(x_{j^\prime}\) in the design. The distance measure is usually taken to be squared Euclidean distance: that is, if \(u=(u_1,u_2,\ldots,u_p)\) then we define
High values of this criterion are desirable.
Additional Comments¶
Note that the resulting design will not truly be optimal with respect to the chosen criterion because only a finite number of LHCs will be generated.